Welschinger Specialization of Generalized Block–Göttsche Polynomials

Definitions and mathematical statement

Let $S$ and $\beta_S$ denote a surface and a curve class in the intended scope of the supplied conjecture. The excerpt denotes its generalized Block--Göttsche polynomial by \[ \operatorname{BPS}_{\beta_S}^{S}(q)\in{\mathbb{Z}}[q,q^{-1}], \] where $q$ is a formal variable, and denotes the corresponding real Welschinger count by $W_{\beta_S}^{S}$. Evaluation of this Laurent polynomial at $q=-1$ is well defined. These descriptions identify the intended invariants but do not supply their complete definitions.

The conjecture attributed in the supplied excerpt to Argüz--Bousseau asks whether, for every admissible pair $(S,\beta_S)$, \[ \operatorname{BPS}_{\beta_S}^{S}(-1)=W_{\beta_S}^{S}. \] Here "admissible" is a placeholder for hypotheses missing from the record, rather than a defined class of surfaces and curve classes. In particular, the record does not specify the real structure on $S$, the admissible curve classes, the incidence conditions, or the conventions defining the Welschinger count. Consequently, this is a conditional transcription of the intended equality, not a fully specified universally quantified conjecture.

The background motivates these polynomials through toric Block--Göttsche Laurent polynomials, described as specializing to genus-zero complex Gromov--Witten counts at $q=1$ and real Welschinger counts at $q=-1$. It proposes an extension using higher-genus Gromov--Witten invariants with an insertion denoted $\lambda_g$, for $g\in{\mathbb{Z}}_{\geq0}$. The excerpt asserts a unique Laurent polynomial characterized by a generating-series identity after $q=e^{iu}$, but that identity is truncated; it cannot serve here as a definition of $\operatorname{BPS}_{\beta_S}^{S}$.

The scope of the supplied partial results is unresolved. The original excerpt asserts the equality for del Pezzo surfaces, phrased there as $n\leq8$, without defining $n$ in the retained passage. The supplied dated literature assessment instead reports a proof for blowups of the projective plane at at most six general points and describes an all-$n$ conjecture as unproved. These assertions are retained as source claims; their hypotheses and relationship require review.

Short English statement

The conjecture attributed to Argüz--Bousseau asks whether evaluating the generalized Block--Göttsche polynomial at $q=-1$ gives the corresponding real Welschinger count. The supplied record omits hypotheses and invariant definitions, and its accounts of the proved range differ. A precise unconditional formulation requires source review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30005941$, exact source problem code catalogue entry, title "Welschinger Specialization of Generalized Block--Göttsche Polynomials." The record gives proposed year 2024 and attributes the conjecture and excerpted theorems to Argüz--Bousseau; its separate proposer field is empty.
• Real and Logarithmic Enumerative Geometry (2024), Oberwolfach Reports, report 23/2024, pp. 1285--1338, DOI 10.4171/owr/2024/23. Supplied report reference and literature-source link: https://doi.org/10.4171/owr/2024/23. The supplied extract contains a truncated defining identity and an asserted del Pezzo result for $n\leq8$.
• BPS polynomials and Welschinger invariants, supplied literature-source link: https://arxiv.org/abs/2506.02770. The supplied assessment attributes to Arguez and Bousseau a proof of the specialization for blowups of the projective plane at at most six general points, while describing the all-$n$ conjecture as unproved.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified here. The record labels the problem partially_solved and the statement corrected_verified, and reports a check against the official Oberwolfach report. These labels and verification claims are supplied metadata, not independent mathematical verification by this draft.